B3: Stochastic modelling and deterministic limit of catalytic surface processes J. Starke1 , C. Reichert2 , M. Eiswirth3 , and K. Oelschl¨ ager4 1
2
3
4
Institute of Applied Mathematics & Interdisciplinary Center for Scientific Computing, University of Heidelberg,
[email protected] Interdisciplinary Center for Scientific Computing, University of Heidelberg,
[email protected] Fritz-Haber-Institute of the Max Planck Society, Berlin,
[email protected] Institute of Applied Mathematics, University of Heidelberg,
[email protected]
Three levels of modelling, microscopic, mesoscopic and macroscopic are discussed for the CO oxidation on low-index platinum single crystal surfaces. The introduced models on the microscopic and mesoscopic level are stochastic while the model on the macroscopic level is deterministic. It can be derived rigorously for low-pressure conditions from the microscopic model, which is characterized as a moderately interacting many-particle system, in the limit as the particle number tends to infinity. Also the mesoscopic model is given by a many-particle system. However, the state space there is a lattice, such that in contrast to the microscopic model the spatial resolution is reduced. The derivation of deterministic limit equations is in correspondence with the successful description of experiments under low-pressure conditions by deterministic reaction-diffusion equations while for intermediate pressures phenomena of stochastic origin can be observed in experiments. The models include a new approach for the platinum phase transition, which allows for a unification of existing models for Pt(100) and Pt(110). The rich nonlinear dynamical behaviour of the macroscopic reaction kinetics is investigated and shows good agreement with low pressure experiments. Furthermore, for intermediate pressures, noise-induced pattern formation, which has not been captured by earlier models, can be reproduced in stochastic simulations with the mesoscopic model.
2
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
1 Introduction Stochastic modelling with many-particle systems and derivation of the corresponding deterministic limit is important in many areas of science. The present work concentrates on the CO oxidation on Pt single crystal surfaces but the techniques presented here can certainly be used in other areas as well. 1.1 CO oxidation on Pt single crystal surfaces Pattern formation under nonequilibrium conditions has been studied using a number of catalytic surface reactions on a large variety of different catalysts [9, 10, 26]. In order to distinguish genuine self-organized patterns from influences of catalyst inhomogeneities, it is necessary to work with uniform surfaces; best suited are oriented single crystal surfaces. The system studied most extensively under these conditions is the CO oxidation on Pt single crystals [24, 8, 26, 37]. In ultra-high vacuum experiments, the elementary processes were elucidated. The reaction proceeds via the classical Langmuir-Hinshelwood mechanism, CO + ? COad O2 + 2? → 2Oad
(1)
COad + Oad → CO2 + 2?, where ? is a vacant adsorption site on the Pt surface. It is important to note that there is asymmetric inhibition of adsorption, i.e. preadsorbed CO blocks oxygen adsorption but not vice versa. In addition, the adsorbate-induced phase transition 1×2 1×1 for Pt(110)
and
(2)
hex 1×1 for Pt(100) has to be taken into account because the surface structure influences the reactivity. The considered experimental situation assumes constant partial pressures of CO and O2 in the well-mixed gaseous phase. The produced CO2 disappears immediately from the Pt surface. It is therefore sufficient to model the adsorbed CO molecules and oxygen atoms on the Pt surface in addition to the platinum phase. On a molecular level, the relevant elementary processes are of stochastic nature. Thus, fluctuations were shown to strongly influence the behavior in experiments with field emitter tips and corresponding Monte-Carlo simulations [42, 41]. In contrast, pattern formation on extended single crystal surfaces at low pressure ( . 10−4 mbar) did not reveal any effects which would suggest a stochastic origin, and could indeed be successfully modelled with (deterministic) reaction-diffusion equations [8].
Stochastic modelling and deterministic limit of catalytic surface processes
3
The reason is that at low pressures there occur due to the diffusion of adsorbed CO on the Pt surface about 106 site changes per adsorption event, i.e. the surface is well mixed on a length scale of about 1µm and fluctuations on the molecular level are averaged out. With increasing pressure smaller and smaller patches can be regarded as well mixed, the size of a critical nucleus decreases [2]. The reaction-diffusion models are expected to fail and stochastic effects can become relevant. An experimental observation at an intermediate oxygen pressure (pO2 = 10−2 mbar) is reproduced in Fig. 1. The CO pressure
(a)
(b)
Fig. 1. (a) Snapshots of a Pt(110) single crystal surface showing the so-called raindrop patterns using EMSI (Ellipso Microscopy for Surface Imaging) [37]. The time between the snapshots is 160ms. The length scale is 100 × 70µm. The partial pressures are pCO = 7 × 10−3 mbar and pO2 = 2.2 × 10−2 mbar. (b) Space-time diagram of the raindrop [37] showing 1.6s × 100µm.
pCO had been stepwise increased to a point shortly before the whole surface would switch to the CO-covered state. CO nuclei were observed to originate at various places, forming a ring-shaped pattern, but were subsequently destroyed (propagation failure). These phenomena are called raindrop patterns due to the similarity to damped out waves on a water surface in starting rain. They seem to appear randomly distributed all over the catalyst surface [38]. Consequently, a unified stochastic model was developed which reproduces the meanfield limit at low pressures, but also describes the stochastic effects observed in experiments at intermediate pressure. In addition for these pressures, the reaction is no longer isothermal because the elevated turnover releases more heat, which is also included in the presented modelling. 1.2 Stochastic modelling and deterministic limit Various types of many-particle models on small scales have been proposed to describe and analyse phenomena in science including fluctuations and other stochastic effects. To analyse the behaviour of those microscopically defined systems on macroscopic scales, it is useful to study the many-particle models in the limit as the particle number tends to infinity. A class of models
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J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
with strong local interaction essentially between immediately neighbouring particles leads to the so-called hydrodynamic limit [39, 5, 28]. On the other hand, models where any particle has a substantial albeit weak influence on essentially all other particles, i.e. models with global interactions lead to the so-called McKean-Vlasov limit [33]. To describe the experimental conditions of the CO oxidation on Pt considered in the present paper the limit behaviour of many-particle models exhibiting a local interaction with moderate interaction range which is much larger than the typical distance between neighbouring particles but much smaller than the total system size has to be studied. This results in a situation inbetween the hydrodynamic and the McKean-Vlasov limit. In the following we present and investigate two types of these stochastic models with local interactions between many mesoscopically neighboring particles, namely a stochastic lattice model, also known as box model or Gillespie model and a moderately interacting many-particle model in continuous space. Monte-Carlo or stochastic cellular automaton models are not considered, as they are usually not able to reproduce with today’s computer simulations the experimentally observed phenomena with realistic parameters. Examples of stochastic lattice models using cells as spatial discretization can be found in [18] and [15]. Some analysis of these models has been based on the Master equation while a mathematically rigorous derivation of the deterministic limit was performed for chemical models assuming either specific reaction kinetics with at most quadratic terms or for systems with only one species specific polynomial kinetics [29, 30]. A rigorous derivation of the deterministic limit for particular many-particle models with moderate interaction, which model the evolution of biological populations, can be found in [34]. Applications of this approach have been employed to model the spatio-temporal pattern formation of myxobacteria [40] and to derive the Smoluchowski equation in the context of an astrophysical application [16]. The present paper is organized as follows: A mesoscopic lattice model for isothermal CO oxidation and its deterministic limit for the spatially homogeneous case are presented in Section 2. In Section 3 a limit theorem for a microscopic model, namely a moderately interacting many-particle system, is presented, which gives a rigorous result of the macroscopic reaction-diffusion equations as deterministic limit equations. The stochastic and thermokinetic effects which come into play at intermediate pressures are modelled in Section 4.
2 Stochastic lattice model for isothermal CO oxidation The presented stochastic lattice model describes CO oxidation on Pt on a mesoscopic level with stochastic birth-death processes for the number of ad-
Stochastic modelling and deterministic limit of catalytic surface processes
5
sorbed CO molecules, oxygen atoms and adsorption sites in (1 × 1) surface structure. 2.1 Model formulation and birth-death processes Instead of distinguishing every single adsorption site like in microscopic Monte-Carlo simulations or in lattice gas approaches, the Pt surface is divided into a two-dimensional array or lattice of cells of mesoscopic size each containing N adsorption sites. Each cell is assumed to be well-mixed. The spatial resolution can thus be varied by changing N which automatically rescales the characteristic length. The state of a cell (i, j) containing N adsorption sites at time t ≥ 0 is described by (i,j) N (t) CO (i,j) (i,j) (i,j) (3) XN (t) = NO (t) ∈ SN = {0, . . . , N }3 , (i,j) N1×1 (t) (i,j)
(i,j)
(i,j)
where NCO , NO , and N1×1 denote for each cell (i, j) the numbers of CO molecules, oxygen atoms, and adsorption sites in a non-reconstructed (1×1) surface structure. The specific labels (i, j) of each cells are only printed for the considered interactions between cells. In order to characterize a stochastic process corresponding to the reaction mechanisms, transition probabilities to other states P XN (t + h) = XN (t) + δXN | XN (t) for small h have to be specified. We introduce the concentrations uN := NNCO , vN := NNO , and wN := NN1×1 ; in the limit N → ∞ they are denoted by u, v, and w. The terms appearing below in the transition probabilities concerning NCO and NO essentially correspond to those used for the (deterministic) reconstruction model for Pt(110) in [31]. The parameters and rate constants are defined in Table 1. For further discussions see [31, 11, 25]. Adsorption • Birth of a CO molecule on the surface. NCO → NCO + 1. P XN (t + h) = XN (t) + (1, 0, 0) | XN (t) = ξ (t) h + o(h) N pCO κCO sCO 1 − uN
• Birth of two adsorbed oxygen atoms. NO → NO + 2. P XN (t + h) = XN (t) + (0, 2, 0) | XN (t) =
1×1 N pO2 21 κO srec O (1 − wN (t)) + sO wN (t) 2
× ((1 − uN (t)) (1 − vN (t))) h + o(h)
(4)
(5)
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J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
By assuming the adsorption rate of CO to be proportional to 1 − uξ independent of the oxygen coverage as in [31], one implicitly drops the conservation constraint imposed on NO , NCO , and the number of vacant sites by the pure Langmuir-Hinshelwood mechanism. On the other hand, it has been observed experimentally that the presence of oxygen on the surface does not noticably influence the adsorption of CO [26, 12]. Consequently, we assume that CO molecules can also be adsorbed at O-covered sites, whereas the dissociative adsorption of an oxygen molecule can only take place at two neighboring free sites. The probability that a particular site is occupied neither by CO nor by O is then (1 − u)(1 − v) instead of 1 − u − v like it is used in [31]. Since the first term is larger only by uv, the difference is negligible if u or v is small (i.e. at low pressure). At intermediate pressure the new term is in better agreement with experiments [36]. The exponent ξ is introduced formally to model a precursor effect in the adsorption of CO as in [31], but it was always set to 1 in the present computations. Note that the sticking coefficient of oxygen on the 1 × 1 phase is higher than on the reconstructed phase which can lead to oscillatory, doubly metastable, and excitable behaviour of the reaction kinetics. Desorption • Death of a CO molecule through desorption. NCO → NCO − 1. P XN (t + h) = XN (t) + (−1, 0, 0) | XN (t) = 1×1 rec N kdes (1 − wN (t)) + kdes wN (t) uN (t)h + o(h)
(6)
The desorption of CO molecules from the reconstructed and the 1×1 phase has been distinguished here because the difference in the binding energies cannot be neglected in the case of Pt(100). Reaction • Death of a CO molecule and an oxygen atom through reaction. NCO → NCO − 1, NO → NO − 1. P XN (t + h) = XN (t) + (−1, −1, 0) | XN (t) = N kre uN (t)vN (t)h + o(h)
(7)
For a well-mixed patch the reaction probability is proportional to the product of the concentrations (mass action kinetics).
Stochastic modelling and deterministic limit of catalytic surface processes
7
Phase transition • Death of an adsorption site in a 1×1 patch and birth of a site in a reconstructed patch. N1×1 → N1×1 − 1. P XN (t + h) = XN (t) + (0, 0, −1) | XN (t) = N krec frec (uN (t), wN (t)) wN (t)h + o(h) (8) • Death of a site in a reconstructed patch and birth of a site in a 1×1 patch. N1×1 → N1×1 + 1. P XN (t + h) = XN (t) + (0, 0, 1) | XN (t) = N k1×1 f1×1 (uN (t), vN (t)) (1 − wN (t)) h + o(h) ,
where
λ
λ
frec(uN , wN ) = (1 − ) (1 − uN ) + (1 − wN ) ,
(9)
(10)
and f1×1 (uN , wN ) = (1 − ) uNλ + wNλ ,
λ > 1 , ∈ [0, 1].
PSfrag replacements
PSfrag replacements
(a)
(11)
(b) weq
weq 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
1
ueq
0
0.2
0.4
0.6
0.8
1
ueq
1×1 rec = kdes , at different = 0.2, λ = 4, kdes Fig. 2. Equilibria of the kinetics for v ≡ 0, kk1rec ×1 values of . (a) = 0.1 (b) = 0.3. Introducing different binding energies for CO on 1×1- and reconstructed phase and thus a coupling between u and w, had almost no effect on the curves.
This ansatz is motivated by the fact that the phase transition proceeds via nucleation and growth. The probability for nucleation is determined solely by the CO coverage, but the growth of a phase can to some extent be autocatalytic which leads to a dependence of the rate of growth on the concentration
8
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
of the phase itself. Therefore we choose e.g. for the growth rate of the 1×1 phase on a reconstructed surface a weighted sum of u and w to some power λ, respectively (cf. eqs. (10) and (11)). A highly nonlinear dependence of this rate on u has been obtained experimentally by Hopkinson et al. [23]. They measured the growth rate of the 1×1 phase on a hex-R reconstructed Pt(100) surface to depend on the CO coverage on the hex-R phase to a power of about 4.5. It is plausible that the exponent for w should be of the same order of magnitude. The reverse transition is modelled in an analogous way. In order not to introduce too many parameters, the same and λ are used, however, the rate constants are allowed to be different. The effect of the weight is shown in Fig. 2. For = 0.1, the equilibrium portion of 1×1 phase can be expressed as a function of ueq , as expected (cf. Fig. 2a). At higher values of both phases are stable within a certain range of ueq (cf. Fig. 2b). Diffusion • Diffusive jump ! processes of a CO molecule (i, j) to neighoring cells. ! in cell (i,j) (i,j) NCO − 1 NCO ∆i ±1 0 with ∈ , . (i+∆i,j+∆j) → (i+∆i,j+∆j) ∆j 0 ±1 NCO NCO +1
P
(i,j)
XN
(i+∆i,j+∆j)
XN
!
(i,j)
|
XN
(i+∆i,j+∆j) XN
(t + h) =
!
−1 0 ! (i,j) 0 XN (t) + (i+∆i,j+∆j) 1 XN 0 0
1 (i,j) (t) = N DCO uN (t) + o(h) 4
(12)
Suitable boundary conditions have to be added. The diffusion of adsorbed oxygen is negligible. Transition probabilities to other states behave like o(h) (It is also assumed that the probability for a transition is zero if it would lead out of the state (i,j) space.). Given an initial distribution for XN (0), there is a Markov jump process in continuous time with state space SN whose transition probabilities satisfy the above equations. Its paths can be chosen right-continuous with left limits. Because the state space SN is bounded, explosions of particle numbers cannot occur, and the process is well-defined for t ∈ [0, ∞) [3]. The time (i,j) (i,j) (i,j) evolution of P XN (t) = Y(i,j) | XN (0) = X0 , Y, X0 ∈ SN , is governed by a master equation [20, 19, 14, 22].
Stochastic modelling and deterministic limit of catalytic surface processes
CO adsorption impingement rate sticking coefficient saturation coverage precursor exponent O2 adsorption impingement rate sticking coefficient
κCO sCO us ξ
Pt(110)
Pt(100)
3.135×105 mbar−1 s−1 1 1 1
2.205×105 mbar−1 s−1 1 1 1
κO 5.858×105 mbar−1 s−1 srec 0.3 O s1O×1 0.6 saturation coverage vs 1 Arrhenius rates ν exp(− kBET ) CO desorption CO desorption reaction phase transition
phase transition exponent weight
3.75×105 mbar−1 s−1 0.001 0.3 1
rec rec kdes νdes = 5×1013 s−1 rec Edes = 32.3 kcal/mol 1×1 1×1 kdes νdes = 5×1013 s−1 1×1 Edes = 32.3 kcal/mol kre νre = 5×105 s−1 Ere = 8.1 kcal/mol k1×1 ν1×1 = 103 s−1 E1×1 = 6.9 kcal/mol krec νrec = 0.2×103 s−1 Erec = 6.9 kcal/mol
rec νdes = 3.323 × 1012 s−1 rec Edes = 27.5 kcal/mol 1×1 νdes = 8.640 × 1014 s−1 1×1 Edes = 35.0 kcal/mol νre = 1.185 × 109 s−1 Ere = 14 kcal/mol ν1×1 = 2.417 × 1011 s−1 E1×1 =25 kcal/mol νrec = 4.833 × 1010 s−1 Erec =25 kcal/mol
λ
4 0.3
4 0.1
9
Table 1. Kinetic parameters for Pt(110) and Pt(100).
2.2 Deterministic Limit To investigate the dynamical properties of the model it is convenient to use the deterministic limit of the introduced many-particle model. The deterministic limit is given for the homogeneous case in the following. We denote the stochastic processes describing the dynamics of the densities by uN (t) 1 (13) xN (t) = XN (t) = vN (t) . N wN (t)
Note that all transition intensities (cf. eqs. (4–9)) are proportional to a product of the system size N and a function of the concentrations only. Because of this scaling property it can be proved rigorously that, if limN →∞ xN (0) = x0 ∈ [0, 1]3 holds with probability one, the paths of the stochastic processes xN (t) approximate the solution x(t) of the initial value problem given by the system of ODEs
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J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
1×1 rec u˙ = pCO κCO sCO 1 − uξ − kdes (1 − w) + kdes w u − kre uv 2 1×1 v˙ = pO2 κO srec (1 − u)(1 − v) − kre uv O (1 − w) + sO w
w˙ = k1×1 f1×1 (u, w)(1 − w) − krec frec (u, w)w,
(14)
and the initial condition x(0) = x0 at large particle numbers in the following sense: For arbitrary t ≥ 0:
lim
sup |xN (s) − x(s)| = 0 with probability one.
N →∞ 0≤s≤t
In addition, a central limit theorem can be shown which states that fluctuations of the concentrations vanish like O √1N . The proofs follow directly from theorems by Kurtz [32, 13]. A non-rigorous approach to these results is known in the physical literature as van Kampen’s system size expansion [14]. Bifurcation analysis Subsequently we present some aspects of the bifurcation structure of system (14) with parameters appropriate for Pt(110) and Pt(100). For a more complete treatment see [36]. The computations were performed using algorithms contained in the AUTO 97 package by Doedel et al. [6]. Abbreviations of the bifurcations found are listed in Table 2, for details see [21, 43, 17]. For Pt(110), bifurcation diagrams in pCO and pO2 have been computed for several fixed sample temperatures. The kinetic parameters used are listed in Table 1. At higher temperatures (500 K–560 K) the bifurcation diagram is mainly organized by a cusp and two Takens-Bogdanov points. In total there are 12 parameter regions with different dynamical behaviour (cf. Fig. 3), but only regions 1–5 are physically relevant because the remaining regions are too small to be detected in typical experiments. In areas 1, 2, and 3 there is only one attractor, a stable node and an asymptotically stable periodic orbit, respectively. The maximal width of the oscillatory region 2 at 540 K amounts to ≈ 10% of the value of pCO at the supercritical Hopf bifurcation. It decreases towards higher as well as lower temperatures. In 4 and 5 two attractors coexist; in 5 there are two stable nodes, while in 4 the system reaches either a stable node or a small asymptotically stable periodic orbit. The width of region 4 is . 1% of pCO at 540 K, so it could possibly be detected experimentally. The areas with nontrivial dynamics, such as bistability or oscillations, move towards higher partial pressures as temperature is increased. Phase portraits of the dynamics in the different parameter regions are sketched in Fig. 3. In the pictures containing three fixed points the lower one can always be identified with a reactive, only partially non-reconstructed surface with a relatively high oxygen coverage (reactive state), the upper one with a mainly CO-covered 1×1 surface (poisoned state). From a physical point of view the model presented here yields almost the same results as the one proposed in ref. [31]. The most important distinction
Stochastic modelling and deterministic limit of catalytic surface processes
11
6 p O2
5
sn
sl 1
SNL
11
NSL
sniper
h
TB2
12
4
DH2
h0
snp
SNL 3
TB1
sl NSL
2
snp
sn
h0 10 7 8
C
9 SNL sniper SNL
6 1
DH1
pCO
1
2
3
4
5
6
7
8
9
10
11
12
Fig. 3. Sketches of the complete bifurcation diagram at 500 K–560 K (top) and phase portraits in different parameter regions (bottom) for Pt(110). Hopf bifurcations, and saddle-node bifurcations involving a stable node and a saddle with a one-dimensional unstable manifold are drawn with solid lines, saddle-node bifurcations involving a saddle with a one-dimensional and another with a two-dimensional unstable manifold are drawn with dash-dotted lines. The dashed curves indicate global bifurcations, cf. Table 2. In the phase portraits stable nodes are represented by filled circles, saddle points with a two-dimensional unstable manifold by empty circles, and saddle points with a one-dimensional unstable manifold by half-filled circles. Asymptotically stable periodic orbits are indicated by solid lines, unstable ones by dashed lines.
is that here the two curves of saddle-node bifurcations do not merge for the investigated parameter region in a second cusp when pCO and pO2 are increased, rather bistability persists even at atmospheric pressures, in agreement with
12
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager Bifurcation Hopf bifurcation (supercritical) Hopf bifurcation (subcritical) saddle-node saddle loop homoclinic bifurcation saddle-node/infinite period saddle-node on a loop saddle node of periodic orbits cusp Takens-Bogdanov bifurcation degenerate Hopf bifurcation saddle-node loop neutral saddle loop trace 0 saddle loop Takens-Bogdanov-cusp
Codimension 1 1 1 1
Abbreviation h h0 sn sl
1
sniper
1 2 2 2 2 2
snp C TB DH SNL NSL
3
TBC
Table 2. Commonly used denotations of bifurcations and their abbreviations. By a degenerate Hopf bifurcation we mean the one described in [17].
experiment. In fact, this is due to the change of the adsorption kinetics of oxygen, as was checked by repeating the computation of the saddle-node curves with the term that was used in former models [31]. As in [31], the dynamics seems to be essentially two-dimensional; for instance a period doubling transition to chaos which was observed experimentally, could not be found. In the case of Pt(100), the difference in the binding energies of CO on 1 × 1- and hex phase must be taken into account. Moreover, some kinetic parameters have to be changed, most importantly the sticking coefficient of oxygen on the hex phase is much smaller than on the 1×2 phase of Pt(110). The weight is increased to = 0.3, in order to model hysteresis in the phase transition; the influence of adsorbed oxygen on the restructuring processes has been neglected. The kinetic parameters used have been adapted from an older model [25]; they are listed in Table 1. For details of the bifurcation structure see [36]. The oscillatory region is much broader than in the case of Pt(110). It compares favourably with experiments (cf. Figs. 4a and 4b). In contrast to earlier models [25, 1] no arbitrary parameters have to be introduced through so-called defect terms to get the required behaviour. In contrast to Pt(110), Pt(100) exhibits two distinct bistable regions. The one at low pressures is due to the hysteresis in the phase transition and extends to zero oxygen pressure, while the one at high pressures is controlled by the reaction dynamics and again does not close. Overall, the bifurcation fine structure turned out somewhat simpler than with Pt(110).
Stochastic modelling and deterministic limit of catalytic surface processes (a)
13
(b)
pO2 [10−5 mbar]
100
h(100)
sl(100)
10
sn(110)
PSfrag replacements
1 1
100
10
pCO [10−6 mbar]
Fig. 4. Occurence of oscillations at 480 K. (a) The curves of Hopf- and sl bifurcations are plotted for Pt(100) using a logarithmic scale. The line on the right is the curve of saddle-node bifurcations corresponding to the loss of stability of the reactive state for Pt(110). (b) Corresponding experimental diagram [11].
3 Limit theorem for many-particle systems in continuous space In this Section, which summarizes [35], the relationship between two different approaches to the mathematical modelling of general chemical systems with intermediate particle interactions, i.e. with microscopically large and macroscopically small range, is investigated. In particular, the rigorous derivation of macroscopic limit equations (reaction-diffusion equations) from microscopic moderately interacting many-particle models is discussed. 3.1 Microscopic and macroscopic modelling of chemical reactions For a large number of complex chemical systems, like the CO oxidation on Pt, the elementary reaction steps are understood, i.e., the microscopic dynamics of the chemical components is known. In contrast to this, many models describing macroscopic phenomena like spatio-temporal pattern formation should be improved, or, at least, their foundation on the microscopic description has to be supplemented. In order to model particular chemical reactions on a microscopic level we consider a class of interacting many-particle systems. Especially, we focus on the use of moderately interacting many-particle systems, where for large particle numbers the mean distance between neighbouring particles is much smaller than the range of the interaction, which in turn is much smaller than the spatial system size. This is fulfilled for many important chemical reactions, where the dynamics is reaction- rather than diffusion-controlled. These systems can
14
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
be regarded as being locally well-mixed. To distinguish the model from the stochastic lattice model described in Section 2 we use the term “microscopic model” for the approach in this section because the positions of individual particles are considered here in continuous space and not on a discrete lattice which corresponds to a reduced spatial resolution. Nevertheless, by using densely populated cells or a moderately scaled interaction, both models are associated with a local averaging in connection with the respective ineraction. Similarly as in [34] we study for any N ∈ N a system of ≈ N particles in Rd . To model different reacting species the set of all particles is divided into a finite collection of K subsets, where K is independent of N . The dynamics of the individual particles is determined by three contributions, namely • immigration, i.e., the entering of new particles into the system e.g. by adsorption of a gaseous species on a surface, • reaction, i.e., the transformation of pairs of interacting particles into particles of another species, and • diffusion. As far as the interaction among the particles is concerned, we suppose similarly as in [34] that immigration- and reaction rates depend on regularized versions of the empirical processes (16) of the various species. For fixed N ∈ N these regularizations are defined as convolutions of the respective empirical processes with a kernel ψN . Additionally, in our model the reaction rates involve some interaction potential φN describing the influence of the distance between the reacting particles, i.e., pair-interaction is included. With δ0 denoting Dirac’s δ-distribution we assume that lim ψN = lim φN = δ0
N →∞
N →∞
in S 0 (Rd ) sufficiently slow.
(15)
The rate of convergence in (15) has to be slow enough, in particular, in order that the various contributions to the interaction scale in a moderate way and hence moderately interacting many-particle systems are obtained, cf. [34]. In contrast to [34], in the model discussed here the diffusion may vanish for certain species. Moreover, no drift is acting on the particles. Last not least, the analysis in [34] is simplified considerably by the absence of explicit pairinteraction. We are aiming at the study of the many-particle systems for large particle numbers, i.e., we investigate the limit N → ∞. Especially, the convergence of the systems of empirical processes to the solution of a system of reactiondiffusion equations will be proved. This limit system represents a macroscopic model of the respective chemical reaction. As an important application the particular example of the heterogeneous catalytic CO oxidation on Pt is considered for low pressures. The already mentioned 106 site changes of an adsorbed CO molecule per adsorption event due to diffusive hopping results in a locally well-mixed system. The asymmetric inhibition as well as the dynamical adsorbat dependent reconstruction of
Stochastic modelling and deterministic limit of catalytic surface processes
15
the Pt structure can fully be considered also in this microscopic moderately interacting many-particle model. The description of this interesting surface reaction with its complicated spatio-temporal dynamical behaviour (cf. Section 2.2 for a numerical bifuraction analysis of the kinetics) is a suitable test for the proposed abstract chemical model. The mathematical modelling in this example including specific rates for the physical processes is presented in Section 3.5. Additionally, the corresponding system of partial differential equations for the macroscopic model is given there. Sections 3.2 and 3.3 contain mathematical details for the general microscopic model and a limit theorem describing its relation to the corresponding macroscopic model. In Section 3.4 some remarks on the proof of the limit theorem can be found. 3.2 Mathematical details of the microscopic model for general reactions r For N ∈ N and t ≥ 0 we use disjoint subsets MN (t), r = 1, ..., K of N to describe the particles in the various subpopulations marked with r. k Let XN (t) ∈ Rd be the position of particle k in the N -th system at time t. For fixed N ∈ N the distribution of the particles in space is characterized by empirical processes
XrN (t) =
1 N
X
δXNk (t) ,
r (t) k∈MN
t ≥ 0, r = 1, . . . , K,
(16)
which obviously take values in the space of nonnegative measures on Rd . The empirical processes will be used for two purposes: • The reaction- and immigration rates describing the interaction between the particles depend on these processes. • They are studied in the limit as N → ∞ to determine the asymptotic behaviour of the many-particle systems. To quantify the interaction the empirical processes XrN are not employed directly. Instead, they are replaced by regularized versions ρrN , which are obtained by convolving XrN with some suitable kernel ψN , i.e., ρrN (., t) = XrN (t) ∗ ψN ,
t ≥ 0, r = 1, . . . , K, N ∈ N.
(17)
We suppose that the family ψN , N ∈ N, is obtained by the scaling 1/d
ψN (x) = θN ψ1 (θN x), where the function ψ1 satisfying Z ψ1 ∈ Cb (Rd ; [0, ∞)), dx ψ1 (x) = 1, Rd
x ∈ Rd , N ∈ N,
ψ1 (−y) = ψ1 (y), y ∈ Rd ,
(18)
(19)
16
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
is fixed. For the scaling coefficients θN , N ∈ N, in (18) we suppose θ1 = 1,
θN ∈ [1, ∞), N ∈ N,
lim θN = ∞ sufficiently slow.
N →∞
(20)
It will be convenient to assemble ρrN , r = 1, . . . , K, in the function ρN : Rd × [0, ∞) → [0, ∞)K defined by ρN (x, t) = ρ1N (x, t), . . . , ρK x ∈ Rd , t ≥ 0. (21) N (x, t) ,
As a further abbreviation we employ ρN (x, y, t) =
1 ρN (x, t) + ρN (y, t) , 2
x, y ∈ Rd , t ≥ 0,
(22)
which will be utilized in the reaction rates.
Remark 1. Obviously, the transition from an empirical process XrN to its regularization ρrN may be considered as a spatial blurring of the point masses describing the positions of the individual particles. If this blurring is strong enough, i.e., if θN → ∞ sufficiently slow, cf. (20), it may be expected that asymptotically as N → ∞ local fluctuations of mass concentrations vanish in ρrN , and therefore ρ = (ρ1 , . . . , ρK ) = limN →∞ ρN should exist under rather general circumstances. Now, for fixed N ∈ N the different contributions to the dynamics of the N -th many-particle system are described in detail. To put our considerations involving random variables and stochastic processes on a firm basis we can work with a suitable filtered probability space (Ω, F, (Ft )t≥0 , P), where (Ft )t≥0 is some filtration satisfying the usual conditions, cf. [27]. We assume that any stochastic process appearing subsequently in this section is adapted to (Ft )t≥0 . Diffusion We suppose that during their lifetimes the particles perform independent Brownian motions, where the diffusion constant is σr ≥ 0, if the respective particle belongs to species r. Since the particles of certain species may be immobile, σr = 0 is allowed. To model the resulting paths of the partik , k ∈ N, may be cles independent standard Rd -valued Brownian motions BN employed. Immigration It is assumed that particles may enter the system with density-dependent rates. More precisely, we suppose the existence of sufficiently regular functions β r ∈ Cb1 Rd × [0, ∞)K ; [0, ∞) , r = 1, . . . , K,
Stochastic modelling and deterministic limit of catalytic surface processes
17
such that P a particle of species r is immigrating in U during (t, t + δ] Ft (23) Z = δN dx β r (x, ρN (x, t)) + o(δ), as δ & 0, U
t ≥ 0, r = 1, . . . , K, U ⊆ Rd measurable.
Reaction In nature various different types of reactions may occur. For notational simplicity we consider in our general model only the cases, where different particles of species r and r 0 can react to turn either into one particle of species p, or two particles of species p and p0 , or three particles of species p, p0 and p00 . Whereas r and r 0 are allowed to coincide, we assume that the species p, p0 and p00 are different. As a consequence, for that type of reactions the species of the reaction products can be completely described by a set B ∈ Θ3K , where Θ3K is the set of all subsets of {1, . . . , K} with 1 ≤ card(B) ≤ 3. To model in the N -th system the influence of the spatial distance between two reacting particles on the reaction rate we use some kernel φN . As in (18) we assume that φN is obtained by a scaling 1/d
φN (x) = τN φ1 (τN x),
x ∈ Rd , N ∈ N,
(24)
from some fixed function φ1 . This function and the sequence τN , N ∈ N are supposed to satisfy similar conditions as in (19) and (20). To take into account catalytic effects we introduce functions K B 2 ; [0, ∞) , r, r0 = 1, . . . , K, B ∈ Θ3K , (25) βr,r 0 ∈ Cb [0, ∞)
B where the set B arising as upper index in βr,r 0 describes the species of the particles appearing after the reaction. As in (23) we then assume particles k 6= l of species r and r 0 located in x and y react Ft P (26) during (t, t + δ] to yield particles of species given by B δ B = φN (x − y)βr,r as δ & 0, 0 ρN (x, y, t) + o(δ), N x, y ∈ Rd , t ≥ 0, r, r0 = 1, . . . , K, B ∈ Θ3K ,
where ρN is introduced in (17), (21) and (22). Moreover, we suppose that all particles resulting from a reaction between particles in x, y ∈ Rd emerge at the same point (x + y)/2. Remark 2. In particular cases it can happen that certain reaction products do not contribute to further reactions, i.e., actually, they leave the system under consideration. This situation would correspond to B = ∅.
18
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
To drop the restriction to at most three particles resulting from any reaction or the assumption that the species of these particles are different we should assume that B is an arbitrary subset or an unordered tuple of elements of {1, . . . , K}. Also splitting reactions, where one particle is divided into several fragments belonging to different species, are possible. For such reactions we may introduce additional reaction rates βrB , r = 1, . . . , K, B a subset or an unordered tuple of elements of {1, . . . , K}, in (25). Apart from some trivial modifications our mathematical considerations in [35] also remain valid for the generalizations mentioned in this remark. 3.3 A limit theorem The relationship between the microscopic model, i.e., the many-particle system described in Section 3.2, and its macroscopic equivalent, i.e., a nonlinear system of partial differential equations, is specified by the subsequent result. Theorem 1. Suppose that suitable regularity conditions are satisfied. Then, lim sup |hXrN (t), f i − hρr (., t), f i| = 0,
N →∞ t≤T ∗
in probability,
(27)
r = 1, . . . , K, f ∈ Cb1 (Rd ) ∩ L2 (Rd ), where the limit densities collected in the [0, ∞)K -valued function ρ = (ρ1 , . . . , ρK ) solve K X X 0 ∂ r B ρ (x, t) = β r (x, ρ(x, t)) − ρr (x, t)ρr (x, t)βr,r 0 (ρ(x, t)) ∂t 0 K
(28)
r =1 B∈Θ3
+
K 1 X 2 0 00
X
r ,r =1 {B∈Θ3K :r∈B}
0
00
ρr (x, t)ρr (x, t)βrB0 ,r00 (ρ(x, t)) +
σr2 ∆ρr (x, t), 2
x ∈ Rd , t ∈ [0, T ∗ ], r = 1, . . . , K. The regularity conditions needed for this theorem are stated in detail in [35]. First, we need various conditions on the scaling coefficients θN , τN , N ∈ N, which appear in (18) and (24), and further coefficients aN , pN , N ∈ N. Next, the interaction kernels, ψ1 , φ1 , cf. (18) and (24), and the kernel W1 , cf. (30) and (31) in Section 3.4, have to be regular enough. Finally, the solution (ρ1 , . . . , ρK ) of the limit equation (28) should be sufficiently smooth in the time interval [0, T ∗ ]. We note that aN , pN , N ∈ N, and W1 are needed for technical reasons, i.e., they have no meaning in the modelling but are employed to deduce the theorem mathematically rigorously.
Stochastic modelling and deterministic limit of catalytic surface processes
19
3.4 Some remarks on the proof of the limit theorem The proof is based on systems of stochastic differential equations hXrN (t), f (., t)i = =
1 N +
X
X
k f (XN (t), t)
(29)
r (t) k∈MN
k f (XN (0), 0)
r (0) k∈MN
1 N
1 − N
+
1 N
Z tZ 0
Z
1 2N
Rd
JrN (dx, ds)f (x, s)
K t X
X
X
0 r 0 =1 r B∈Θ3K k∈MN (s)
Z
t
K X
0 r 0 ,r 00 =1
k ρB N ;r,r 0 ;k,l (ds)f (XN (s), s)
0
r l∈MN (s)
X
X
ρB N ;r 0 ,r 00 ;l,l0 (ds)f
{B∈Θ3K :r∈B} l∈M r0 (s) N 00
X l (s)+X l0 (s) N N ,s 2
r l0 ∈MN (s)
Z t X σr2 k ∆f (XN (s), s) ds 2N 0 r (s) k∈MN Z σr t X k k + ∇f (XN (s), s) · BN (ds) N 0 r (s) k∈MN Z X ∂ 1 t k + , ds f (XN (s), u) N 0 ∂u u=s r +
k∈MN (s)
t ≥ 0, r = 1, . . . , K, f ∈ Cb2,1 (Rd × [0, ∞)), N ∈ N,
for the empirical processes to describe the dynamics of the many-particle systems. In (29) we denote for r = 1, . . . , K and N ∈ N by JrN some random field in Rd × [0, ∞), which describes the instants in space-time, where in the N -particle system particles of species r are immigrating. Moreover, for any r r0 N ∈ N, r, r0 = 1, . . . , K, k ∈ MN (.), l ∈ MN (.) and B ∈ Θ3K the process B ρN ;r,r0 ;k,l marks that instant, where the particles k and l in the r-th and the r0 -th species, respectively, interact to produce particles described by the set B. As a first application of (29) some formal arguments can be used to guess the limit dynamics (28). In this context, we suppose that for any t ≥ 0 the empirical processes X1N (t), . . . , XK N (t) indeed converge to limits having smooth densities ρ1 (·, t), . . . , ρK (·, t). Next, we represent the stochastic integrals on the right side of (29) as sums of a martingale and a process of bounded variation of first order. We then observe that the martingales vanish as N → ∞ and
20
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
take into account (15). As a formal result, (29) turns into a weak version of (28). Now, the empirical processes X1N , . . . , XK N and their supposed respective limits ρ1 , . . . , ρK can be compared directly. For that purpose we introduce with K X QN (t) = k(XrN (t) − ρr (., t)) ∗ WN k22 , t ≥ 0, N ∈ N, (30) r=1
a family of stochastic processes, which describe for fixed N some L2 -distance between the N -particle system and the limit ρ. In (30) we use kernels 1/d
WN (x) = aN W1 (aN x),
x ∈ Rd , N ∈ N,
(31)
where W1 is a sufficiently regular probability density and aN % ∞ as N → ∞. To investigate the asymptotics as N → ∞ we employ (28) and (29) to deduce for any QN , N ∈ N, a stochastic differential equation describing its time evolution. After extensive, but fairly straightforward estimates of the various terms of this stochastic differential equation essentially Gronwalls Lemma can be used to verify that QN vanishes as N → ∞. In particular, lim E 1 ∧ rN sup QN (t) = 0 (32) N →∞
t≤T ∗
for some suitable sequence rN , N ∈ N, with limN →∞ rN = ∞ is obtained. Finally, (27) is a simple consequence of (32). For the details of the proof of the theorem we refer to [35]. 3.5 Application to heterogeneous catalysis: CO oxidation on Pt As indicated in the introduction the general considerations on the modelling of chemical reactions with many-particle models in continuous space is applied here to the example of the CO oxidation on low-index Pt-surfaces in a low-pressure atmosphere, where the partial pressures of O2 and CO are kept constant. In the present context it is particularly important that for modelling purposes the use of moderately interacting many-particle systems is justified, as under the considered conditions a locally well-mixed system is given. The description in Section 3.2 is translated into a moderately interacting many-particle system in continuous space R2 with three species, i.e., d = 2 and K = 3. For the N -particle system we may employ X1N , X2N and X3N for the empirical processes associated to the CO-molecules, O-atoms and Pt-1×1 adsorption sites, respectively. As far as the Langmuir-Hinshelwood mechanism and the adsorbate induced Pt phase transition given in (1) and (2) are concerned, some particular subsets of physical processes are modelled in terms of immigration, namely the adsorption of CO and O2 and the transition from Ptrec to Pt-1×1. The corresponding immigration rates are β 1 , β 2,2 and β 3 . Since
Stochastic modelling and deterministic limit of catalytic surface processes
21
the empirical processes related to oxygen describe O-atoms, the adsorption of O2 -molecules is related to another trivial extension of our model introduced in Section 3.2, namely the simultaneous immigration of two particles at the same place. For this reason the notation β 2,2 instead of β 2 is employed. The remaining reactions in (1) and (2), i.e., the desorption of CO, the creation of CO2 and the transition from Pt-1×1 to Pt-rec, are described as reactions according to Section 3.2. For those cases, after taking into account in particular ∅ Remark 2, the reaction rates β1∅ , β1,2 and β3∅ have to be specified. To quantify the above-mentioned rates we use the considerations in Section 2 which is based on [36]. In particular, after some slight modifications we choose
β
β 1 (x, ρ1 , ρ2 , ρ3 ) = pCO κCO sCO H1 (1 − (ρ1 )ξ )S(x),
2,2
1
2
3
(x, ρ , ρ , ρ ) =
1×1 3 pO2 κO (sO ρ
+
srec O H1 (1
(33a)
3
− ρ ))
(33b)
1
2
2
(H1 (1 − ρ )H1 (1 − ρ )) S(x)/2,
β 3 (x, ρ1 , ρ2 , ρ3 ) = k1×1 f1×1 (ρ1 , ρ3 )H1 (1 − ρ3 )S(x), β1∅ (ρ1 , ρ2 , ρ3 ) ∅ β1,2 (ρ1 , ρ2 , ρ3 )
β3∅ (ρ1 , ρ2 , ρ3 )
=
1×1 3 kdes ρ
+
rec kdes H1 (1
= kre , 1
3
= krec frec (ρ , ρ ),
(33c)
3
− ρ ), 2
(33d) (33e)
1
2
3
x ∈ R , ρ , ρ , ρ ∈ [0, ∞).
(33f)
1×1 1×1 rec The kinetic parameters κCO , sCO , ξ, κO , sO , srec O , k1×1 , kdes , kdes , kre and krec , which are strictly positive, are given in Table 1. pCO and pO2 denote the partial pressure of CO and O2 , respectively, in the gaseous phase above the Pt-surface. The function f1×1 defined in (10) is used with the densities ρ1 , ρ2 , ρ3 which results in
f1×1 (ρ1 , ρ3 ) = (1 − )(ρ1 )λ + (ρ3 )λ ,
ρ1 , ρ3 ∈ [0, ∞),
(34a)
whereas for frec we have frec(ρ1 , ρ3 ) = (1 − )(H1 (1 − ρ1 ))λ + (H1 (1 − ρ3 ))λ ,
ρ1 , ρ3 ∈ [0, ∞). (34b)
In (34) the constants ∈ [0, 1] and λ > 1 have to be chosen suitably, as it is described in Section 2. The function S(x) with spatial variable x in (33a) - (33c) is used to model the size of the Pt-surface, which is finite. We suppose ( 1, if |x| ≤ R, 1 d S ∈ Cb (R ; [0, 1]) with S(x) = 0, if |x| ≥ R0 , for suitable constants R0 > R > 0. In order to avoid negative rates in (33) and (34) the function [0, 1] 3 u → 1 − u from Section 2 is replaced by [0, ∞) 3 u → H1 (1 − u), where H1 (v) = max{v, 0}, v ∈ R.
22
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
In [36], only the spatially homogeneous situation was considered. In particular, the diffusive hopping of adsorbed CO-molecules mentioned in Section 3.1 is neglected. Here, we model that phenomenon in terms of independent Brow√ nian motions acting on the CO-molecules with diffusion constant σ1 = 2D for some D > 0. Both adsorbed O-atoms and Pt-1×1 adsorption sites are considered as immobile, i.e., σ2 = σ3 = 0 is supposed. To complete our mathematical model of this particular chemical reaction we still have to choose suitable kernels ψ1 and φ1 and associated scaling coefficients θN , N ∈ N, and τN , N ∈ N. Since present knowledge of the chemistry does not seem to impose restrictions, it suffices to take into account the regularity conditions mentioned in the theorem in Section 3.3. By the particular form of the rates in (33) and (34) we obtain as limit dynamics the system ∂ 1 ρ = pCO κCO sCO (1 − (ρ1 )ξ )S (35a) ∂t 1×1 3 rec − (kdes ρ + kdes (1 − ρ3 ))ρ1 − kre ρ1 ρ2 + D∆ρ1 , ∂ 2 1×1 3 3 1 2 2 1 2 ρ = pO2 κO (sO ρ + srec O (1 − ρ ))((1 − ρ )(1 − ρ )) S − kre ρ ρ , (35b) ∂t ∂ 3 ρ = k1×1 f1×1 (ρ1 , ρ3 )(1 − ρ3 )S − krec frec (ρ1 , ρ3 )ρ3 . (35c) ∂t We note that the function H1 does not appear in (35) as a consequence of a maximum principle, which implies 0 ≤ ρ1 (x, t), ρ2 (x, t), ρ3 (x, t) ≤ 1,
x ∈ R2 , t ≥ 0,
if this condition is imposed for the initial time t = 0. The obtained reaction-diffusion equation (35) can now be used as microscopically justified macroscopic model for subsequent analysis and numerical simulation. While this model is in excellent agreement with experiments over a wide parameter range, it will fail when stochastic effects become important, as discussed in the next section.
4 Stochastic and thermokinetic effects in CO oxidation For increasing pressure, there are fewer and fewer diffusion sites changes of adsorbed species per adsorption event. Therefore the characteristic length over which the system can be regarded as well mixed decreases until the corresponding number of sites becomes so small that fluctuations have to be taken into account and the mean-field limit is not applicable any more. Additionally, at intermediate pressures, the system is no longer isothermal due to heat release by the reaction. The temperature T can be modelled using
Stochastic modelling and deterministic limit of catalytic surface processes
23
the heat balance equation [4] for a volume V of the catalyst with heat capacity C: CO O CV T˙ = QCO ad + Qdes + Qad + Qre + Qrad + Qcond .
(36)
O CO QCO ad , Qad , Qdes , and Qre are the rates of production of heat energy through the reaction processes, Qrad and Qcond are the balances for the gain and loss of heat through radiation, conduction and the feedback system regulating the temperature during the experiments by electrical heating. Linearising the equation by Taylor expansion around the temperature T0 0 in the absence of any reaction and rescaling the temperature to θ = T −T T0 leads to the equation CO CO θ˙ = γ (Had κCO pCO sCO (1 − uξ ) − Hdes kdes (θ)u
2 1 O2 1×1 κO pO2 (srec + Had O (1 − w) + sO w) (1 − u)(1 − v) 2 +Hre kre (θ)uv − µθ + Dθ ∇2 θ.
(37)
Here γ represents a scaled heat capacity, the parameter µ contains the effectivity of the feedback control: ρs Cρb lT T0 4σes (T0 )T03 + λh /lT . µ(T0 ) = Cρb lT γ(T0 ) =
(38) (39)
The parameters entering these equations are summarized in Table 3. The full deterministic model is obtained by the combination of equation (37) for the rescaled temperature with the reaction-diffusion equations u˙ = κCO pCO sCO 1 − uξ − kdes (θ)u − kre (θ)uv +DCO (θ)∇2 u srec O (1
v˙ = κO pO2 −kre (θ)uv
(40)
− w) +
s1×1 O w
(1 − u)(1 − v)
2
w˙ = k1×1 (θ)f1×1 (u, w)(1 − w) − krec (θ)frec (u, w)w.
(41) (42)
and suitable boundary conditions. This macroscopic model (40) – (42) resulted in Section 3.5 in the interior of the bounded Pt surface (i.e. for S(x) = 1) as limit (35) from the moderately interacting many-particle model using temθ perature dependent Arrhenius rates k(θ) = ν exp(− kBET0 ) exp( kBET0 1+θ ) = E ν exp(− kB T ). For large µ and not too high reaction rates the system would remain isothermal so that thermal effects can be analyzed using µ as bifurcation
24
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager CO Had heat of adsorption of CO
135 kJ mol−1
CO Hdes heat loss by desorption of CO
CO ≈ Had
Hre
heat of reaction
20 - 30 kJ mol−1
O2 Had
heat of adsorption of an oxygen atom 115 kJ mol−1
ρs
density of surface atoms
8.84 × 1018 atoms/m2
σ
Stefan-Boltzmann constant
5.6705 × 10−8 Jm−2 K−1 s−1
es
integral emissivity of platinum
0.05 − 0.1
λh
heat conduction coefficient
72 W m−1 K−1
lT
characteristic thickness of surface layer 5 × 10−5 m
C
specific heat capacity of platinum
130 J kg−1 K−1
ρb
bulk mass density of platinum
21.09 × 103 kgm−3
Dθ
diffusion constant of temperature
λh /(Cρb )
Table 3. Parameters required for heat balance equation.
parameter. Thermokinetic effects are a consequence of the asymmetric inhibition of adsorption (i.e. preadsorbed CO blocks oxygen adsorption, but not vice versa) and the strong temperature dependence of CO desorption. Thus an O-covered surface exhibits a high reaction rate and therefore becomes hot while a high CO-coverage keeps the catalyst relatively cool. Since in turn lower T favors CO coverage through reduced desorption, the effect is autocatalytic. A partial bifurcation analysis of the model is reproduced in Fig. 5. For PSfrag replacements
(a) replacements PSfrag
ueq 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0030
(b)
µ 25.
sn
20.
3 4
sn
15.
h
10. 5.
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100 pCO
0. 0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
pCO Fig. 5. (a) Continuation of CO coverage in pCO for thermokinetic model. There is a Hopf bifurcation point (h) between two saddle-node bifurcation points (sn). (b) Continuation of Hopf and sn bifurcations in pCO and µ.
large µ the O-covered surface is excitable close to the Hopf bifurcation and stable pulses exist. With decreasing µ this bifurcation shifts to higher pCO , so
Stochastic modelling and deterministic limit of catalytic surface processes
25
that the O-covered branch becomes subexcitable. Pulses can still be triggered, but they shrink and eventually vanish (Fig. 6). Physically this can be readily
(a)
u
θ
t
x
x
t
(b)
u
θ
t x
x t
Fig. 6. Simulation of pulse propagation using thermokinetic model equations (40) – (42) and (37) in one dimension with rescaled temperature and no-flux boundary conditions. The space variable x is given in µm while the time t is given in s. A CO nucleus was put in the initial conditions on an otherwise predominantly Ocovered surface. (a) For µ = 103 pulses form, propagate but finally die due to rising temperature. (b) For µ = 10 pulses can still be formed, but die quickly. pCO = 5.0 × 10−3 mbar, pO2 = 1.55 × 10−2 mbar, DCO = 10−12 m2 /s, T0 = 520 K, other parameters as in Tables 1 and 3. In the simulation shown heat conductance was chosen unrealistically small in order to visualize where heat production takes place. The effect of nucleation and propagation failure however persists even for a realistic heat diffusion (i.e. very large thermal conductivity).
explained by the temperature effects of the changes in reaction rate. The rate drops sharply on the (predominantly) CO covered areas, because here oxygen adsorption is blocked. However, behind the pulses the reconstruction has been lifted and the reactivity is increased to values even higher than on the original O-covered 1 × 2 surface because of the enhanced sticking coefficient
26
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
of oxygen on the 1 × 1 surface. Therefore, the temperature locally rises to values higher than at the beginning. Since the temperature spreads rapidly via heat conductance, the CO pulses are overrun from the inside (a hotter surface cannot maintain a high CO coverage due to enhanced desorption). The stochastic model described in Section 2 was combined with thermokinetic effects using equation (37). Since heat conductance is much larger than diffusion, it was treated deterministically. For small pressures, the model behaved practically isothermally and followed the deterministic path very closely. In particular no spontaneous nucleation in the bistable or excitable region was observed. In contrast for parameters realistic for intermediate pres2 sure (N = 103 , hopping rate corresponding to DCO = 1.4 × 10−14 ms ) significant fluctuations become visible and critical nuclei did form spantaneously. A computer simulation for pO2 = 10−2 mbar of nucleation, pulse formation and propagation failure is reproduced in Fig. 7. the rate of such events turned out to be extremely sensitive with respect to pCO . The nucleation was obviously random, the subsequent pulse annihilation occurred not too far from the deterministic path (cf. Fig. 6), although relatively small random variations in the maximum local CO coverage led to noticeable variations in the time it took to return to the initial state. The homogeneous nucleation rate on an ideal surface can be roughly estimated using large-deviations analysis [7]. For an equilibrium CO coverage u ¯, a critical coverage u∗ and a critical nucleus size containing n sites, the system cr ∗ 1−¯u 1−u∗ ncr u ¯ u spends a time fraction of u∗ in a state with u ≥ u∗ . 1−u∗
u and u∗ can be taken from the nullclines of the deterministic model, ncr decreases linearly with the pressure and is between 103 and 104 at 10−2 mbar [2]. The expected number of events can be obtained by multiplying with the density of adsorption sites and dividing by the characteristic time, for which we chose the time required for the impingement of one monolayer CO. The resulting function is obviously very sensitive to ncr but also depends crucially on u∗ − u ¯. Reasonable values [37] of 1 to 100 mm−2 s−1 were obtained with ncr = 5000, u∗ = 0.35 and 0.31 < u ¯ < 0.33. Increasing ncr by 2 orders of magnitude (which would correspond to a pressure decrease by 2 orders) resulted in values of about 10−35 mm−2 s−1 which would certainly not be observable in experiment nor computer simulations. Obviously, surface inhomogeneities always play a role on real catalysts. Nevertheless, the experimental observations and their close correspondence to a realistic model clearly suggest that at least a significant fraction of the observed nuclei form uniformly distributed over the surface. The presented effect therefore constitutes the first example of mesoscopic pattern formation (50 – 100µm) in a surface reaction that is initiated by microscopic fluctuations and cannot be fully captured in a deterministic description.
Stochastic modelling and deterministic limit of catalytic surface processes
27
Fig. 7. Twodimensional simulation of raindrop patterns on a predominantly Ocovered Pt(110) surface using stochastic model with included thermokinetic effects. CO nucleation occurred due to coverage fluctuations. For the stochastic simulation, 200 × 200 cells, N = 103 adsorption sites for each cell and pCO = 5.22 × 10−3 mbar, pO2 = 1.55 × 10−2 mbar, DCO = 1.4 × 10−14 m2 /s, T0 = 520 K, µ = 103 were used.
5 Conclusion Spatio-temporal pattern formation in the CO oxidation on Pt has been studied experimentally over a wide range of parameters. The observed phenomena
28
J. Starke, C. Reichert, M. Eiswirth, and K. Oelschl¨ ager
mostly appear deterministically, except for very small catalyst areas or at sufficiently high pressure where also temperature variations become observable. In order to develop a consistent description of the large variety of experimental patterns, a unified stochastic model of CO oxidation including thermal effects for all low-index plane Pt surfaces was developed, which approaches a realistic deterministic limit when the surface can be regarded as well-mixed, but also reproduces stochastic effects with increasing pressure. It includes earlier models as limiting cases, i.e. approaches Monte-Carlo simulations for small cells and a reaction-diffusion system for intermediate particle interaction, and also reproduces stochastic effects, such as random nucleation.
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